We provide novel reduced-order observer designs for continuous-time nonlinear systems with measurement error. Our first result applies to systems with continuous output measurements, and provides observers that converge in a fixed finite time that is independent of the initial state when the measurement error is zero. Our second result applies under discrete measurements, and provides observers that converge asymptotically with a rate of convergence that is proportional to the negative of the logarithm of the size of a sampling interval. Our observers satisfy an enhanced input-to-state stability property with respect to the measurement error, in which an overshoot term only depends on a recent history of the measurement error. We illustrate our observers using a model of a single-link robotic manipulator coupled to a DC motor with a nonrigid joint, and in a pendulum example.